Question

In a triangle \(\mathrm{ABC}\), If \(\mathrm{a}=5, \mathrm{~b}=4\) and \(\mathrm{A}=60^{\circ}\), then \(\mathrm{c}\) is a root of the equation (a) \(x^{2}-4 x-9=0\) (b) \(x^{2}+4 x+9=0\) (c) \(x^{2}-4 x+9=0\) (d) \(x^{2}+4 x-9=0\)

Short answer

Question: Determine the equation which has side length c as its root, given a triangle ABC with side lengths a=5, b=4, and angle A=60°. Answer: (a) \(x^{2}-4 x-9=0\)

Step-by-step solution

Apply Law of Cosines

Recall the Law of Cosines: \(c^{2}=a^{2}+b^{2}-2ab\cos{A}\). Plug in the given values: \(c^{2}=5^{2}+4^{2}-2\times5\times4\cos{60^{\circ}}\)

Evaluate the expression

Compute the value of \(\cos{60^{\circ}}\). Then calculate \(c^{2}\): \(c^{2}=25+16-40\times\frac{1}{2} =25+16-20=21\)

Find the value of c

Take the square root of both sides: \(c=\sqrt{21}\)

Determine the equation with c as the root

Test equations (a) to (d) to see which one has \(\sqrt{21}\) as its root. The correct answer is (a): \((\sqrt{21})^{2}-4(\sqrt{21})-9=0\) The answer is (a) \(x^{2}-4 x-9=0\).

Key concepts

Understanding Triangle Properties

When studying triangles, there are several fundamental properties one must understand. A triangle consists of three sides and three angles, and the sum of the angles always equals 180 degrees. One remarkable attribute of triangles is that the lengths of the sides and the measures of the angles are interconnected. For instance, in a right triangle, the Pythagorean theorem provides a direct relationship between the lengths of the sides. However, what about non-right triangles?

That's where the Law of Cosines becomes crucial. This powerful extension of the Pythagorean theorem applies to any type of triangle, not just right triangles. It connects the length of one side of a triangle to the lengths of the other two sides and the cosine of the included angle. In symbols, for a triangle with sides of lengths a, b, and c, and an angle A opposite side a, the Law of Cosines says that: \(c^2 = a^2 + b^2 - 2ab\text{cos}(A)\).

Special Cases of Triangles

Equilateral triangles, where all sides and angles are the same, and isosceles triangles, where two sides and angles are the same, showcase specific consistent properties that simplify calculations. In our exercise, we deal with a general triangle which requires the Law of Cosines to solve for the unknown side. This underlines the significance of mastering a variety of triangle properties for tackling geometry problems.

Trigonometric Functions and Their Applications

Trigonometric functions are the backbone of solving many geometric problems, especially those involving triangles. These functions include sine, cosine, and tangent, among others. Specifically, the cosine function, which we use in the Law of Cosines, relates the angle of a triangle to the ratio of the adjacent side to the hypotenuse—in a right triangle.

However, trigonometric functions also extend their usefulness to non-right triangles, as our exercise demonstrates. The cosine of an angle, denoted as \(\text{cos}(A)\), can take on values between -1 and 1, and is a crucial part of the Law of Cosines. It's key to note that these functions are periodic and have specific values for commonly used angles, like \(\text{cos}(60^{\text{o}}) = \frac{1}{2}\).

Understanding the Unit Circle

A clear way to comprehend trigonometric functions like cosine is to reference the unit circle, where the angle in question corresponds to a point on the circle’s circumference. The x-coordinate of this point is the cosine of the angle, which helps in visualizing why the cosine of 60 degrees is 0.5. Grasping these concepts allows students to transition smoothly from understanding right triangles to tackling more complex geometric problems involving various types of triangles.

Solving Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^2 + bx + c = 0\), where a, b, and c are constants, and \(a eq 0\). These equations are fundamental in algebra and appear frequently in various branches of mathematics, including geometry, as seen in our original problem.

To solve a quadratic equation, you can use a variety of methods such as factoring, completing the square, or applying the quadratic formula. Knowing how to solve these equations is essential since they can represent a wide array of practical problems, including the calculation of areas, optimization problems, and, as in the given example, finding the lengths of sides in geometry when applied in combination with the Law of Cosines.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true. In our exercise, once we calculated the length of side \(c\), we tested it against different quadratic equations to see which one had \(c\) as its root. This reinforces the importance of recognizing how quadratic equations can serve as the bridge between algebra and geometry, and understanding their solutions is crucial for students at all levels of mathematics education.